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B. LINE PERPENDICULAR TO A PLANE

A. PERPENDICULAR LINES

Definition: (perpendicular lines)

Two lines a and b are perpendicular to each other if the angle between them is 90.

For intersecting lines we can easily define the perpendicularity. But, if the lines are skew lines then we will take any point on one of the lines and draw a parallel line to the other line. If the angle between these two intersecting lines is 90 then the given skew lines are said to be perpendicular.

Theorem:If one of two parallel lines is perpendicular to a third line, the other is perpendicular too.

Proof:

Let m and b be two parallel lines and m be perpendicular to c (Figure 1.36).

 

Through any point A, let us draw lines m1 and c1 so that m1 // m and c1 // c.

 

Since m c, the angle between m1 and c1 is 90.

 

On the other hand, since m1 // m and m // b, we get m1 // b.

 

So the angle between b and c is also 90.

 

That means b and c are perpendicular lines

Example 26:By using perpendicularity, prove that if two lines are parallel to the same line they are parallel.

Solution:Let m, n and d be three lines. Let

m // n and d // n. We need to prove that m // d.

 

Let A be a point on n. We can draw a plane α perpendicular to n at A. So n ⊥ α.

 

Since m // n and n ⊥ α, m ⊥ α because of the previous theorem.

 

Since d // n and n ⊥ α then d ⊥ α .

 

Since m and d are perpendicular to α by using the previous theorem they are parallel.

Check Yourself 8

1.State the followings as true or false.

a.Two lines d and k are parallel to the same plane. If a line l is perpendicular to line d it must be always perpendicular to line k.

b.Two perpendicular lines are given in space. If a line is perpendicular to these lines at their intersection point then this line is perpendicular to the plane, which includes the perpendicular lines.

c.Given that d, k and l are three lines in space. If d and k are perpendicular to line l they must be parallel to each other.

d.Two lines d and k in a plane are perpendicular to the same line l. Another line m is parallel to line l then m is perpendicular to d and k, also.

Answers

1.a) False b) True c) False d) True

B. LINE PERPENDICULAR TO A PLANE

Definition: (line perpendicular to a plane)

A line is said to be perpendicular to a plane if it is perpendicular to every line in this plane.

 

 

Let us say that m is a line and α is a plane. If it is given that m ^ α then m is perpendicular to any line in α.

If m ^ α then m intersects α. To prove this statement let us assume that m does not intersect α.

 

In this case there are two possibilities for m and α :

1.m is in α. Then since it is not perpendicular to a line in α, that is itself, m is not perpendicular to α.

2.m is parallel to α. In this case in α there can be found a line parallel to m. So m can not be perpendicular to α.



 

In both conditions m is not perpendicular to α.

 

Therefore, m intersects α.

Definition: (inclined line)

If a line intersects a plane but not perpendicular to the plane it is called an inclined line.

Theorems:

1. If a line is perpendicular to two intersecting lines lying in a plane then it is perpendicular to the plane.

2.Through any given point in space, there can be drawn one and only one plane perpendicular to a given line.

3.If one of two parallel lines is perpendicular to a plane then the other line is also perpendicular to the same plane.

4.Two lines perpendicular to the same plane are parallel.

5.Through a point in space, there can be drawn only one line perpendicular to a given plane.

6.If a line is perpendicular to one of two parallel planes, it is perpendicular to the other.

Proofs:

 

1.We need to prove that if a line is perpendicular to two intersecting lines in a plane it is perpendicular to any line in this plane.

Let d be a line perpendicular to two lines m and n lying in α. Let A be the intersection point of m and n.

 

It is obvious that d is perpendicular to every line in α which is parallel to either one of m or n (Figure 1.37).

 

So we should check for the lines which are not parallel to neither m nor n.

 

Let x be any line intersecting both m and n.

 

We have to prove that d is perpendicular to x too.

 

Let us shift lines d and x so that A is on d and x.

 

Let c be any line in α intersecting m, n, x at

points C, D, E respectively.

 

On line d let us take two points B and B' so that BA = B'A.

 

Then DBAC and DB'AC are congruent, similarly DBAD and DB'AD are congruent (by S.A.S.).

 

So BD = B'D and BC = B'C.

 

Then DBDC and DB'DC are congruent (by S.S.S.). That means ∠BDC = ∠B'DC.

 

Then DBDE and DB'DE are congruent triangles (by S.A.S.).

 

So BE = B'E and DBAE and DB'AE are congruent (by S.S.S.).

 

Hence ∠BAE = ∠B'AE = 90.

 

So d is perpendicular to x.

 

Therefore d is perpendicular to any line in α. So d ⊥ α.

2.We have two cases:


Date: 2015-12-11; view: 787


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